RETROSPECT 263

Peering Past the Frontier

As we move on to other systems of Z-equations, we ﬁnd that

particular examples tend to be less and less of interest. Do you

really care exactly how many solutions to the system

33x

2

− 42xy + z

72

= 445

x

2

− 34xyz + z

333

= 12

x + y

2

+ 2z

3

= 11

there are with x, y, and z all integers? Unless this problem has

some burning practical signiﬁcance, it is not very interesting

merely on its own. We just made it up at random. On the other

hand, it does represent the kind of problem that interests us in

general, namely, ﬁnding the solutions to a system of Z-equations.

Any adequate theory we develop should eventually be able to tell

us about this random system and all other systems too.

Thus, our focus tends to shift to whole classes of problems, and

therefore to the structural or qualitative statements we can make

or prove about them. For example, what computable properties of

a system of Z-equations can guarantee that there are only ﬁnitely

many solutions with the variables taking on only rational values?

Such questions, which are at the frontier of research, lead back to

more structural questions about varieties and the absolute Galois

group of Q.

For another example, consider the statement: The absolute

Galois group of Q is big. Now, make it into a precise assertion and

prove it. That might include proving that for any positive integer n

and any prime p, there is a Galois representation whose image

2

is

all of GL(n, F

p

). (It is not known if this is true.) But there will be

many other ways of envisioning the “bigness” of the absolute Galois

group of Q, according to various other aspects of its structure.

In this way, after we leave behind particular Z-equations or sys-

tems that have provoked our study in the ﬁrst place, the structures

2

The image of a group representation φ is the set of all elements in the target that are

actually values of φ.

264 CHAPTER 23

themselves—the absolute Galois group of Q, for example—tend to

take center stage. As the structures become more understood, par-

ticular problems—the congruent number problem, for example—

will achieve their solution. This dialectic between the general and

the speciﬁc is very common in the history of mathematics.

Thus, the focus of theoretical interest in the study of systems

of Z-equations tends to shift, as the theory is developed, to Galois

groups and their representations. We begin to ask questions: How

big can representations of the Absolute Galois group of Q be? How

can certain families of such representations be parametrized? But if

for any reason we need to say something about a particular system

of Z-equations, the theory should stay closely enough connected

to the motivating Diophantine problems to be a powerful tool for

attacking these problems.

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